Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution $\mathrm{EPD}_λ(μ,σ)$, including tests for normality and Laplacity when $λ$ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two $Z$-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain $Z$-scores that are nearly independent and distributed as standard Gaussians, resulting in a $χ_2^2$ distribution valid for any sample size (up to very high precision for $n\geq 20$). The case $λ=1$ leads to a powerful test of fit for the Laplace($μ,σ$) distribution, whose empirical power is superior to all $39$ competitors in the literature, over a wide range of $400$ alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.